The following is my personal (i.e., includes all of my mathematical prejudices) ranked list of subject area journals in number theory.
From best to worst:
1) Algebra and Number Theory
2) International Journal of Number Theory
3) Journal de Theorie des Nombres de Bordeaux
4) Journal of Number Theory
5) Acta Arithmetica
6) Integers: The Journal of Combinatorial Number Theory
7) Journal of Integer Sequences
8) JP Journal of Algebra and Number Theory
For a slightly longer list, see
http://www.numbertheory.org/ntw/N6.html
but I don't have any personal experience with the journals listed there but not above.
Moreover, I think 1) is clearly the best (a very good journal), then 2)-5) are of roughly similar quality (all quite solid), then 6) and 7) have some nice papers and also some papers which I find not so interesting, novel and/or correct; I have not seen an interesting paper published in 8).
But I don't think that even 1) is as prestigious as the top subject journals in certain other areas, e.g. JDG or GAFA. There are some other excellent journals which, although not subject area journals, seem to be rather partial to number theory, e.g. Crelle, Math. Annalen, Compositio Math.
Finally, as far as analytic and combinatorial number theory goes, I think 4) and 5) should be reversed. (Were I an analytic number theorist, this would have caused me to rank 5) higher than 4) overall.)
Most journals in math allow you to publish a version of the paper which was previously posted to the arxiv.org. They ask you often to take the copyright for the published version which just slightly differs from the arxiv version. So there is not much difference between having it public or having a slightly different version public. Some journals, on the other hand are free anyway and forever in their public versions, e.g. Theory and application of categories. If you choose a journal carefully you solve most of your concern. Some publishers are notorious of being nasty, expensive, proprietory, nonresponsive to author needs etc. You do not want to publish in expensive envelopes of crap, like Elsevier's Chaos, solitons and fractals used to be.
Best Answer
Let me try to summarize this long discussion in the comments. There are many free resources.
arXiv. It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, for example in Europe (see 5 below).
NSF depository. NSF's stated policy is that all results of NSF-sponsored research "must be available to the public at most in 1 year since their publication". I checked: the site is somewhat confusing but it works.
Many journals are freely available, and many more make their papers available after some time (usually 4-5 years). When you choose a journal to publish your paper, take this into account! Here is a convenient catalog of online journals. I am sure many other libraries have similar catalogs; I find this one convenient. The journals with free access are marked green, with partial access yellow and red. Access may depend on your location or on the date of publication.
Finally there are "pirate" sites. Some of them may have huge collections, larger than many university libraries. They frequently change their names and location. Some keywords may be "bookfi" or "genesis" for books, and "sci-hub" for journal articles. (Some of them may be illegal in some countries).
A simple search on Google and especially on Google Scholar sometimes finds what you need. It could be a place you do not expect. Some saved/cached copy. Some preprint depository that you do not know. Some personal web site, etc.
The very important resource is MathScinet, which unfortunately has no free version. But its German competitor Zentralblatt Math is partially free. Whatever you search there, it gives you only 3 items for free. But by clever choice of search criteria you can obtain amazing results. For really old items, there is also Jahrbuch which is free (it is a subset of Zentralblatt). For new papers, Google Scholar is excellent, especially if you know the author's name and title of the paper. It also sometimes finds you a free copy when available.
EDIT. I asked NSF, and they explained that all NSF-supported papers older than 1 year are really available, though the site is somewhat confusing. One has to click on the title of the paper, and then on a little square which says "pdf".
EDIT 2: I collected some links to free resources on my web page.